Existence and Region of Critical Probabilities in Bootstrap Percolation on Inhomogeneous Periodic Trees

  • Milan BradonjićEmail author
  • Stephan Wagner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10088)


Bootstrap percolation is a growth model inspired by cellular automata. At the initial time \(t=0\), the bootstrap percolation process starts from an initial random configuration of active vertices on a given graph, and proceeds deterministically so that a node becomes active at time \(t=1,2,\dots \) if sufficiently many of its neighbors are already active at the previous time \(t-1\). In the most basic model, all vertices have the same initial probability of being active in the initial configuration. One of the main questions is to determine the percolation threshold (if it exists) with the property that all nodes in the given graph become active asymptotically almost surely (a.a.s.) for the initial probability above this threshold, while this is not the case below the threshold. In this work, we study a scenario where the nodes do not all receive the same probabilities, but to keep the problem tractable, we impose conditions on the shape of the graph and the initial probabilities. Specifically, we consider infinite periodic trees, in which the degrees and initial probabilities of nodes on a path from the root node are periodic, with a given periodicity. Instead of the simple percolation threshold, we now obtain an entire region of possible probabilities for which all nodes in the tree become a.a.s. active. We show: (i) that the unit cube, as the support of the initial probabilities, can be partitioned into two regions, denoted by \(W_0\) and \(\overline{W}_0\), such that the tree becomes (does not become) a.a.s. fully active for any initial probability vector that belongs to \(\overline{W}_0\) (resp. \(W_0\)); (ii) for every node in the tree, we provide the probability that the node becomes eventually active, for any initial probability vector that belongs to \(W_0\); (iii) further, we specify the boundary of \(W_0\) and show how it can be numerically computed.


Cellular Automaton Percolation Threshold Initial Configuration Initial Probability Activation Threshold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Aizenman, M., Lebowitz, J.L.: Metastability effects in bootstrap percolation. J. Phys. A: Math. Gen. 21(19), 3801–3813 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amini, H.: Bootstrap percolation and diffusion in random graphs with given vertex degrees. Electron. J. Comb. 17, 1–20 (2010). #R25MathSciNetzbMATHGoogle Scholar
  3. 3.
    Amini, H., Fountoulakis, N.: What I tell you three times is true: bootstrap percolation in small worlds. In: Proceedings of Internet and Network Economics - 8th International Workshop, WINE 2012, Liverpool, UK, 10–12 December 2012, pp. 462–474 (2012)Google Scholar
  4. 4.
    Amini, H., Fountoulakis, N.: Bootstrap percolation in power-law random graphs. J. Stat. Phys. 155(1), 72–92 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Amini, H., Fountoulakis, N., Panagiotou, K.: Bootstrap percolation in inhomogeneous random graphs. arXiv:1402.2815
  6. 6.
    Balogh, J., Bollobás, B., Duminil-Copin, H., Morris, R.: The sharp threshold for bootstrap percolation in all dimensions. Trans. Am. Math. Soc. 364(5), 2667–2701 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Balogh, J., Bollobás, B., Morris, R.: Majority bootstrap percolation on the hypercube. Comb. Probab. Comput. 18(1–2), 17–51 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Balogh, J., Bollobás, B., Morris, R.: Bootstrap percolation in high dimensions. Comb. Probab. Comput. 19(5–6), 643–692 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Balogh, J., Peres, Y., Pete, G.: Bootstrap percolation on infinite trees and non-amenable groups. Comb. Probab. Comput. 15(5), 715–730 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Balogh, J., Pittel, B.: Bootstrap percolation on the random regular graph. Random Struct. Algorithms 30(1–2), 257–286 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Biskup, M., Schonmann, R.H.: Metastable behavior for bootstrap percolation on regular trees. J. Stat. Phys. 136(4), 667–676 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bollobás, B., Gunderson, K., Holmgren, C., Janson, S., Przykucki, M.: Bootstrap percolation on Galton-Watson trees. Electron. J. Probab. 19(13), 1–27 (2014)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Bradonjić, M., Saniee, I.: Bootstrap percolation on random geometric graphs. Probab. Eng. Inf. Sci. 28(2), 169–181 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bradonjić, M., Saniee, I.: Bootstrap percolation on periodic trees. In: Proceedings of 12th Workshop on Analytic Algorithmics and Combinatorics, ANALCO 2015, San Diego, CA, USA, 4 January 2015, pp. 89–96 (2015)Google Scholar
  15. 15.
    Chalupa, J., Leath, P.L., Reich, G.R.: Bootstrap percolation on a Bethe lattice. J. Phys. C 12, L31 (1979)CrossRefGoogle Scholar
  16. 16.
    Duminil-Copin, H., Van Enter, A.C.D.: Sharp metastability threshold for an anisotropic bootstrap percolation model. Ann. Probab. 41(3A), 1218–1242 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fontes, L., Schonmann, R.: Bootstrap percolation on homogeneous trees has 2 phase transitions. J. Stat. Phys. 132(5), 839–861 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gravner, J., Holroyd, A.E., Morris, R.: A sharper threshold for bootstrap percolation in two dimensions. Probab. Theor. Relat. Fields 153(1–2), 1–23 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Holroyd, A.E.: Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theor. Relat. Fields 125, 195–224 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Janson, S., Luczak, T., Turova, T., Vallier, T.: Bootstrap percolation on the random graph \(G_{n, p}\). Ann. Appl. Probab 22(5), 1989–2047 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Schonmann, R.: Critical points of two-dimensional bootstrap percolation-like cellular automata. J. Stat. Phys. 58(5–6), 1239–1244 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schonmann, R.H.: On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab. 20(1), 174–193 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    van Enter, A., Adler, J., Duarte, J.: Finite-size effects for some bootstrap percolation models. J. Stat. Phys. 60(3–4), 323–332 (1990)MathSciNetCrossRefGoogle Scholar
  24. 24.
    van Enter, A., Fey, A.: Metastability thresholds for anisotropic bootstrap percolation in three dimensions. J. Stat. Phys. 147(1), 97–112 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    van Enter, A., Hulshof, T.: Finite-size effects for anisotropic bootstrap percolation: logarithmic corrections. J. Stat. Phys. 128(6), 1383–1389 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Mathematics of SystemsNokia Bell LabsMurray HillUSA
  2. 2.Department of Mathematical SciencesStellenbosch UniversityStellenboschSouth Africa

Personalised recommendations